Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates

Claus Vogl
(Submitted on 8 Sep 2014)

The evolution of the allelic proportion x of a biallelic locus subject to the forces of mutation and drift is investigated in a diffusion model, assuming small scaled mutation rates. The overall scaled mutation rate is parametrized with θ=(μ1+μ0)N and the ratio of mutation rates with α=μ1/(μ1+μ0)=1−β. The equilibrium density of this process is beta with parameters αθ and βθ. Away from equilibrium, the transition density can be expanded into a series of modified Jacobi polynomials. If the scaled mutation rates are small, i.e., θ≪1, it may be assumed that polymorphism derives from mutations at the boundaries. A model, where the interior dynamics conform to the pure drift diffusion model and the mutations are entering from the boundaries is derived. In equilibrium, the density of the proportion of polymorphic alleles, \ie\ x within the polymorphic region [1/N,1−1/N], is αβθ(1x+11−x)=αβθx(1−x), while the mutation bias α influences the proportion of monomorphic alleles at 0 and 1. Analogous to the expansion with modified Jacobi polynomials, a series expansion of the transition density is derived, which is connected to Kimura’s well known solution of the pure drift model using Gegenbauer polynomials. Two temporal and two spatial regions are separated. The eigenvectors representing the spatial component within the polymorphic region depend neither on the on the scaled mutation rate θ nor on the mutation bias α. Therefore parameter changes, e.g., growing or shrinking populations or changes in the mutation bias, can be modeled relatively easily, without the change of the eigenfunctions necessary for the series expansion with Jacobi polynomials.

The general recombination equation in continuous time and its solution

Ellen Baake, Michael Baake, Majid Salamat
(Submitted on 4 Sep 2014)

The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and solve the latter recursively for generic sets of parameters. We also briefly discuss the singular cases, and how to extend the solution to this situation.

Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

Ute Lenz, Sandra Kluth, Ellen Baake, Anton Wakolbinger
(Submitted on 2 Sep 2014)

In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let x denote today’s frequency of the beneficial type, and given x, let h(x) be the probability that, among all individuals of today’s population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for h(x). We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides interest in its own right, this construction allows a transparent derivation of the series coefficients of h(x) and gives them a probabilistic meaning.

Continuous and Discontinuous Phase Transitions in Quantitative Genetics: the role of stabilizing selective pressure

Annalisa Fierro, Sergio Cocozza, Antonella Monticelli, Giovanni Scala, Gennaro Miele
(Submitted on 2 Sep 2014)

By using the tools of statistical mechanics, we have analyzed the evolution of a population of N diploid hermaphrodites in random mating regime. The population evolves under the effect of drift, selective pressure in form of viability on an additive polygenic trait, and mutation. The analysis allows to determine a phase diagram in the plane of mutation rate and strength of selection. The involved pattern of phase transitions is characterized by a line of critical points for weak selective pressure (smaller than a threshold), whereas discontinuous phase transitions characterized by metastable hysteresis are observed for strong selective pressure. A finite size scaling analysis suggests the analogy between our system and the mean field Ising model for selective pressure approaching the threshold from weaker values. In this framework, the mutation rate, which allows the system to explore the accessible microscopic states, is the parameter controlling the transition from large heterozygosity (disordered phase) to small heterozygosity (ordered one).

Fixation in large populations: a continuous view of a discrete problem

Fabio A. C. C. Chalub, Max O. Souza
(Submitted on 27 Aug 2014)

We study fixation in large, but finite populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, which includes most classical evolutionary processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. Indeed, in the derivation three regimes naturally appear: selection-driven, balanced, and quasi-neutral — the latter two require WS, while the former can appear with or without WS. From the continuous approximations, we then obtain asymptotic approximations for evolutions with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the Evolutionary Stable Strategy (ESS) is outside a small interval around $\sfrac{1}{2}$, the fixation is of dominance type. We also show that outside of the weak-selection regime the dynamics of large populations can have very little resemblance to the infinite population case. In addition, we also show results for the case of two equilibria. Finally, we present a continuous restatement of the definition of an ESSN strategy, that is valid for large populations. We then present two applications of this restatement: we obtain a definition valid in the quasi-neutral regime that recovers the one-third law under linear fitness and, as a generalisation, we introduce the concept of critical-frequency.

Emergent speciation by multiple Dobzhansky-Muller incompatibilities

Tiago Paixão, Kevin E. Bassler, Ricardo B. R. Azevedo
doi: http://dx.doi.org/10.1101/008268

The Dobzhansky-Muller model posits that incompatibilities between alleles at different loci cause speciation. However, it is known that if the alleles involved in a Dobzhansky-Muller incompatibility (DMI) between two loci are neutral, the resulting reproductive isolation cannot be maintained in the presence of either mutation or gene flow. Here we propose that speciation can emerge through the collective effects of multiple neutral DMIs that cannot, individually, cause speciation-a mechanism we call emergent speciation. We investigate emergent speciation using a haploid neutral network model with recombination. We find that certain combinations of multiple neutral DMIs can lead to speciation. Complex DMIs and high recombination rate between the DMI loci facilitate emergent speciation. These conditions are likely to occur in nature. We conclude that the interaction between DMIs may be a root cause of the origin of species.

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model

Yuri S. Semenov, Artem S. Novozhilov
(Submitted on 19 Aug 2014)

We reformulate the eigenvalue problem for the selection–mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples and theoretical findings are compared with numerical calculations.

Can the site-frequency spectrum distinguish exponential population growth from multiple-merger coalescents?

Matthias Birkner, Jochen Blath, Bjarki Eldon, Fabian Freund
doi: http://dx.doi.org/10.1101/007690

The ability of the site-frequency spectrum (SFS) to reflect the particularities of gene genealogies exhibiting multiple mergers of ancestral lines as opposed to those obtained in the presence of exponential population growth is our focus. An excess of singletons is a well-known characteristic of both population growth and multiple mergers. Other aspects of the SFS, in particular the weight of the right tail, are, however, affected in specific ways by the two model classes. Using minimum-distance statistics, and an approximate likelihood method, our estimates of statistical power indicate that exponential growth can indeed be distinguished from multiple merger coalescents, even for moderate sample size, if the number of segregating sites is high enough. Additionally, we use a normalised version of the SFS as a summary statistic in an approximate bayesian computation (ABC) approach to distinguish multiple mergers from exponential population growth. The ABC approach gives further positive evidence as to the general eligibility of the SFS to distinguish between the different histories, but also reveals that suitable weighing of parts of the SFS can improve the distinction ability. The important issue of the difference in timescales between different coalescent processes (and their implications for the scaling of mutation parameters) is also discussed.

Reproductive isolation of hybrid populations driven by genetic incompatibilities

Molly Schumer, Rongfeng Cui, Gil G Rosenthal, Peter Andolfatto
doi: http://dx.doi.org/10.1101/007518

Despite its role in homogenizing populations, hybridization has also been proposed as a means to generate new species. The conceptual basis for this idea is that hybridization can result in novel phenotypes through recombination between the parental genomes, allowing a hybrid population to occupy ecological niches unavailable to parental species. A key feature of these models is that these novel phenotypes ecologically isolate hybrid populations from parental populations, precipitating speciation. Here we present an alternative model of the evolution of reproductive isolation in hybrid populations that occurs as a simple consequence of selection against incompatibilities. Unlike previous models, our model does not require small population sizes, the availability of new niches for hybrids or ecological or sexual selection on hybrid traits. We show that reproductive isolation between hybrids and parents evolves frequently and rapidly under this model, even in the presence of ongoing migration with parental species and strong selection against hybrids. Our model predicts that multiple distinct hybrid species can emerge from replicate hybrid populations formed from the same parental species, potentially generating patterns of species diversity and relatedness that mimic adaptive radiations.

Clonal interference and Muller’s ratchet in spatial habitats
Jakub Otwinowski, Joachim Krug
(Submitted on 18 Feb 2013 (v1), last revised 23 Jul 2014 (this version, v3))

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller’s ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio Ud/s2 between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using again an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.